2 Answers
2

This is discussed briefly as a generalization of Shapiro's cyclic sum
inequality by J. Michael Steele in his book
The Cauchy-Schwarz Master Class.
He remarks that (1.) holds for $n\ge25$ and refers to this paper:
P. J. Bushell, Shapiro’s “Cyclic Sums",
Bull. L.M.S. (1994) 26, 564–574.

This is discussed in detail on the following MathWorld article: Shapiro's Cyclic Sum Constant
.
Detailed proofs of the main result (inequality holds only for even $n \le 12$ and odd $n \le 23$) can be found in the following note by Khrabrov.